Math 4 part 1

B. 1. 2. 3. 4.C. 1. π 2. 30° 3. 60° 4. 45° 5. π , 5π 6 3Lesson 3 2. 110° 3. 454° A. 1. 417° 4. 185° 5. 470° 6. 110° 7. 460° 30′ 8. 567° 55′ 9. 545° 08′ 10. 769° 28′ 45″ B. 1. Ql , 84° 2. Qlll, 35° 3. Q lV, 5° 4. QlV, 5°5. Q lll, 70° 6. Q1, 60° 7. Qll, 10° 8. Q ll, 20° 9. Q1, 29°52′ 10. Q lll, 24° 45′What have you learned1. No 2. 2π rad 3. 20° 34. 100° 5. -21π rad 6. s = 4 cm7. Q1 8. -960° 9. 1152°10. 26.18 cm 25

Module 1 Exponential Functions What this module is about This module is about Exponential Functions. You will learn how to classifyrelations which deals on Interest on bank accounts, radioactive decay, biologicalgrowth, and the spread of infectious diseases. These are examples by which theamount of change depends on the amount of materials present. You will alsodevelop skills in graphing different types of exponential functions. What you are expected to learn This module is designed for you to: 1. Identify certain relationships in real life which are exponential (e.g. population, growth over time, growth of bacteria over time, etc.) 2. Given a table of ordered pairs, state whether the trend is exponential or not. 3. Draw the graph of an exponential function f(x) = ax and describe some properties of the function or its graph. • a>1 • 0<a<1 4. Given the graph of an exponential function determine the : • domain • range • intercepts • trend • asymptote 5. Draw the graph of an exponential function f(x) = Bax and f(x) = ax + c and compare it to the graph of f(x) = ax

How much do you knowAnswer the following:1. Which of the statements is best modeled by an exponential growth? a. The cost of pencils as a function of the number of pencils. b. The distance when a stone is dropped as a function of time. c. The distance of a swinging pendulum bob from the center as a function of time. d. The compound interest of an amount as a function of time.2. Identify the relation which describes an exponential relation from amongthe given table of values.a. c.x -2 -1 0 1 2 x1234 5y1 3 579 y 2 4 9 16 25b. d. x -2 -1 0 1 2 y5 2 125 x0123 4 y 1 3 9 27 813. The graph of a function of the form y = ax passes through which of the following points? a. (-I, 0) b. (1,0) c. (0,1) d. (0,-1)Given the function y = 3(4x) – 2, what is the: 4. y-intercept 5. Trend of the graph 6. Asymptote7. Which of the following functions has a steeper graph? a. y = 2x b. y = -2x c. y = 1 x 2 d. y = 3 x8. Which of the following is a decreasing function? a. y = 4x + 2 b. y = 3(2x) c. y = 3( 4 x) 3 2

d. y = 2( 1 x) 3 9. What is the range of f(x) = 2x – 3? a. y > -3 b. y > - 3 c. y < -3 d. y < -3 10. In function y = 2(4x) – 3, the graph is asymptotic to a. y > 2 b. x –axis c. y = -3 d. y < -3 What you will do Lesson 1Identify Relationships in Real Life Which are Exponential in Nature A rabbit at maturity gives birth to two rabbits. If a rabbit gives birth to 2rabbits, and two rabbits in turn gives birth to two rabbits each, how many rabbitswill there be after four rabbits give birth?Let’s see this in a table:No. of births 1234No. of rabbits 2 4 8 16 This example is exponential in nature and is said to be an increasingfunction or an exponential growth . Relational statements such as mass of a 200 gram sample of an elementbeing reduced to 100 grams after 10 years is an example of a decreasingfunction or an exponential decay. The period where this occurs is called half-life. 3

You will encounter terms such as appreciate and growth for increase, andterms such as depreciate, half-life or decay for decrease in the discussions onexponential functions.Try this outTell whether the following statements describes an exponential growth or decay. 1. The population of Kuhala Island doubles every 3 decades. 2. The amount of radioactive isotope of carbon has a half-life of 5500 years. 3. A colony of bacteria grows by 20% every half hour. 4. An amount deposited by Jessie in a bank earns a compound interest of 6% yearly. 5. The value of a car depreciates 10% of its amount every year. 6. A person deciding to go on a diet for 3 months loses 1 kg of his weight 8 every month. 7. A chain letter where each persons sends 3 letters to 3 persons. 8. A radioactive substance decays after a certain time t. 9. The value of a jewelry appreciates 8% every 5 years. 10. A certain culture of bacteria grows from 500 to 400 bacteria every 1.5 hours. Lesson 2 Given a Table of Ordered Pairs, State Whether the Trend is Exponential or Not A relation which exhibits an exponential change can be described by a setof values in a table. In your study of the table of values of linear and quadraticfunctions, you get a constant value in the first differences in y in a linear functionswhile a constant in the second differences in a quadratic function.Examples:a. y = 3x + 2 x -2 -1 0 1 2 3 4 y = f(x) -4 -1 2 5 8 11 14 3 3 3 3 3 3 First differences in y 4

Notice that a constant value of 3 was obtained in the first differences iny. The relation y = 3x + 2 is a linear function.b. f(x) = – 2x2 + 5 x -3 -1 13579y = f(x) -13 3 3 -13 -45 -93 -157 16 0 -16 -32 -48 -64 First differences in y -16 -16 -16 -16 -16 Second differences in y Notice that a constant value of -16 was obtained in the seconddifferences in y. The relation f(x) = – 2x2 + 5 is a quadratic function. You have seen the behavior of the values of y in the two functions forequal differences in x. Now it is time for you to us study the behavior of thevalues of y for equal differences in x in the third function.c. y = 2x x 012345y = f(x) 1 2 4 8 16 32 22 2 2 2 Equal ratios in yWhen you divide the consecutive values of y you get equal ratios. This type of relation where a constant ratio between two consecutivevalues for y for equal differences in x is what we call an exponential function.Try this outTell whether the following table of values describe a linear, quadratic orexponential function.1. x 123456 y = f(x) 1 3 6 10 15 212. x 012345 y = f(x) 1 3 9 27 81 243 5

3. x 012345 y = f(x) 1 5 9 13 17 214. x 012345 y = f(x) 1 1 1 111 2 4 8 16 325. x 012345 y = f(x) 4 8 16 32 64 1286. x 012345 y = f(x) 5 7 9 11 13 157. x -3 -2 -1 0 1 2 y = f(x) 1 6 9 10 9 68. x 123456 y = f(x) 4 8 16 32 64 1289. x 012345 y = f(x) -3 -1 1 3 5 710. 345 x 012 5 13 29 y = f(x) -2 -1 1 6

Lesson 3Draw the graph of an exponential function f(x) = ax, where a >1 An exponential function of the form y = ax where a >1. To understand fullyexponential function of this form , let us complete the table of values, plot thepoints and graph the function.Examples: 1. Construct a table of values for f(x) = 2x and graph in the coordinate plane. Table of values x -3 -2 -1 0 1 2 3y = f(x) 1 1 1 1 2 4 8 84 2The graph of f(x) = 2x Y 9 8 7 6 5 4 3 2 1 0 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3 One property of the graph is that it passes the point (0, 1) or the graph has itsy – intercept = 1. 2. Construct a table of values for f(x) = 4x and graph in the coordinate plane.x -3 -2 -1 0 1 2 3y = f(x) 1 1 1 1 4 16 64 64 8 4 7

The graph of f(x) = 4x Y 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1X0 -4 -3 -2 -1 -1 0 1 2 3 -2 -3Like the graph of f(x) = 2x, the graph passes (0, 1) and its y – Intercept = 1. To further explore the properties of the graphs, let us plot the points of f(x) =2x and f(x) = 4x on one coordinate plane.Now let us compare the table of values of the two functions. x f(x) = 2x f(x) = 4x -3 1 1 8 64 1 -2 1 16 4 1 -1 1 4 2 1 4 01 12 24 16 38 64What have you noticed? The set of values for y in f(x) = 4x tends to be greater than that of f(x) = 2xas x increases and tends too become much smaller as x decreases. 8

As x becomes smaller, y also becomes too small that it tends to approachzero. But will a value of x give us a value of 0 for y? Think by analyzing thevalues of y. Yes, you are right it will never give us a zero value for y.Compare too the graphs of f(x) = 2x and f(x) = 4x Y f(x) = 4x f(x) = 2x 17 12 3 45 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1X0 -4 -3 -2 -1 -1 0 -2 -3What have you noticed? The graph of f(x) = 2x is wider compared to the graph of f(x) = 4x,meaning the graph of f(x) = 4x is steeper. Now you can draw a conclusion for this. The base of the two functions are4 and 2. This illustrates that the bigger is the base, the steeper is the graph. Notice also that the graph approaches zero as the value of x becomessmaller. From the table of values, it will never get a zero value for y. Meaning,the graph will not touch the x – axis or the line y = 0. We shall now call this linethe asymptote. Let us draw more properties by observing the two graphs. We shall seethe trend of the graph. Always start from the left side of the graph. See that thegraphs increases from left to right. The trend of the graphs of the two functions istherefore increasing. This two functions are increasing functions. The domain (values of x) of an exponential function is the set of realnumbers. The range can be observed from the graph. The graph approacheszero but never touches it. Let us make then zero the boundary or the graph isfrom y > 0 going up. Therefore, the range is the set of all y’s greater than 0 or{y/y > 0}. 9

Summarizing the properties in a table.Function Domain Range Y- Asymptote Trendf(x) = 2x {x/x є R} {y/y > 0} intercept y=0 increasingf(x) = 4x {x/x є R} {y/y > 0} increasing y=1 y=1 y=0 You can now conclude that exponential functions of the form f(x) = ax,where a >1 have these properties. Now it is your turn to try your skills in constructing tables, graphing andanalyzing properties of exponential functions of the form you have learned.Try this outA. Fill the table of values below. Sketch the graph the functions in onecoordinate plane. Analyze and arrange the properties in the table provided. 1. y = 1.5x 2. y = 2.5x 3. y = 3xTable of values x y = 1.5x y = 2.5x y = 3x -3 -2 -1 0 1 2 3 The Properties Range Y- Asymptote Trend Function Domain intercept1. y = 1.5x2. y = 2.5x3. y = 3x 10

B. For each pair of functions, which has a steeper graph? 1. y = 1.5x and y = 3x 2. y = 2x and y = 2.5x 3. y = 3x and y = 3.5x 4. y = 4x and y = 3x 5. y = 4.2x and y = 4x 6. y = 7x and y = 9x 7. y = 5x and y = 1.5x 8. y = 3x and y = 8x 9. y = 6x and y = .6x 10. y = 4x and y = 5x Lesson 4Draw the graph of an exponential function f(x) = ax, where 0< a <1. Now let us try graphing exponential functions of the form f(x) = ax, where is0< a <1. They simply are the base that are fractions between 0 and 1. See howthe graphs differ with that of the previous lesson.Example: Graph y = 1 x 2Table of values The graph x y= 1x Y 9 2 8 -3 8 7 -2 4 6 -1 2 5 01 11 4 2 3 21 2 4 1 31 0X 8 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3 11

Properties of y = 1 x 2 Function Domain Range Y- Asymptote Trend {x/x є R} {y/y > 0} intercept y=0 decreasingy= 1x y=1 2Now, compare the graphs of y = 2x and y = 1 x 2 Table of values x y = 2x y = 1 x 2 -3 1 8 8 -2 1 4 4 -1 1 2 2 01 1 12 1 24 38 2 1 4 1 8The graphs of y = 2x and y = 1 x 2 Y 9 8 7 y = 2x 6 y= 1x 5 4 2 3 2 1X0 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3The graph of y = 1 x is the mirror image of y = 2x with respect to the y - axis. 2 12

The properties of the two functions: Function Domain Range Y- Asymptote Trend {x/x є R} {y/y > 0} intercept y=0 increasingy = 2x {x/x є R} {y/y > 0}y= 1x y=1 2 y=1 y=0 decreasingNote: f(x) = 1 x and f(x)=2-x are the same. 2Try this outA. Complete the table of values. Sketch the graph of each pair of functions in one coordinate plane.1. If y = 3x and y = 1 x 3Table of values x y = 3x y=1x -3 3 -2 -1 0 1 2 32. If y = 4x and y = 1 x 4Table of valuesx y = 4x y=1x-3 4-2-10123 13

B. Tell which of the following is an increasing or decreasing function.1. y = 6x2. y = 4-x3. y = 3.5x4. y = 10x5. y = 8-x6. y = 7-x7. y = 9x8. y = 8x9. y = 12x10. y = 12-xC. Write the properties of the functions in the table:Function Domain Range y- Asymptote Trend intercept1. f(x)= 7-x2. f(x) = 7x3. f(x) = 1 x 54. f(x) = 5-x5. f(x) = 4 x 36. f(x) = 2 x 37. f(x) = 9x8. f(x) = 9-x9. f(x) = 3 x 210.f(x) = 3 -x 2 14

Lesson 5Draw the graph of an exponential function f(x) = Bax and f(x) = ax + c, and compare it to the graph of f(x) = ax If the function is multiplied or added to a constant, how will it affect thereference function f(x) = ax? Let’s find out. Consider the following examples.Example: Sketch the graph of f(x) = 3(2x) and f(x) = 2xTable of values x f(x) = 2x f(x) = 3(2x) -3 1 3 8 8 3 -2 1 4 4 -1 1 3 2 2 01 3 12 6 24 12 38 24The graph of f(x) = 3(2x) and f(x) = 2x 26 Y 25 24 23 f(x) = 3(2x) 22 f(x) = 2x 21 20 1234 19 18 17 16 15 14 13 12 11 109 8 7 6 54 3 2X 1 -01-4 -3 -2 -1 -2 0 -3 15

Check the table of values. You will notice that the values of f(x) = 3(2x) is3 times the values of f(x) = 2x. The graph of f(x) = 3(2x) is translated 3 unitsvertically upwards from the graph of f(x) = 2x. In this case, the y – intercept of f(x)= 3(2x) is 3.Let’s summarize the properties of the two graphs. Function Domain Range Y- Asymptote Trendf(x) = 2x {x/x є R} {y/y > 0} intercept y=0 increasingf(x) = 3(2x) {x/x є R} {y/y > 0} increasing y=1 y=3 y=0 If a constant c is added to f(x) = ax to be transformed into f(x) = ax + c, whatwill happen to the function? You can find out by doing the same procedure as wedid in the first example.Example: Graph f(x) = 2x and f(x) = 2x – 3 x f(x) = 2x f(x) = 2x – 3 -3 1 - 23 8 8Table of values -2 1 - 11 4 4 -1 1 -5 2 2 01 -2 12 -1 24 1 38 5 Y 9 8 f(x) = 2x 7 6 5The graph 4 3 2 f(x) = 2x – 3 1 X0 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3 16

Check the table of values. You will notice that the values of f(x) = 2x - 3 is 3units less than the values of f(x) = 2x. The graph of f(x) = 2x - 3 is translated 3units vertically downwards from the graph of f(x) = 2x. In this case, the asymptoteof f(x) = 2x - 3 is -3.Let’s summarize the properties. Function Domain Range Y- Asymptote Trend1. f(x) = 2x {x/x є R} {y/y > 0} intercept2. f(x) = 2x - 3 {x/x є R} {y/y > 0} y=1 y = 0 increasing y=2 y = -3 decreasingTry this outA. Complete the table of values and graph of the following functions. 1. y = 2(2x) x y = 2(2x) -3 -2 -1 0 1 2 3 2. f(x) = 3x - 4 x f(x) = 3x - 4 -3 -2 -1 0 1 2 3B. Without graphing determine the y-intercept of each function. 1. y = 8x 2. y = 5-x 3. y = 2(6-x) 17

4. y = 3(2x) 5. y = 5( 1 x) 5 6. f(x) = 3(3x) –1 7. f(x) = 2(3x) + 2 8. f(x) = 5( 2 x) + 3 5 9. f(x) = 2x + 3 10. f(x) = 2(3x) – 4C. Without graphing determine the asymptote of each function. 1. f(x) = 9x 2. y = 3(2x) 3. y = 2(7-x) 4. f(x) = 8-x 5. y = 1 x 5 6. y = 5(2x) – 2 7. f(x) = 2(3x) + 3 8. y = 6( 1 x) – 5 2 9. y = 3( 2 x) + 2 3 10. f(x) = 2(5x) – 3 Let’s Summarize 1. For functions of the form y = ax , where a > 1 • y = ax is an increasing function • The asymptote of the curve is the x-axis or line y = 0. • The domain is the set of real number. • The range is y /y > 0. • The greater is the base the steeper is the graph. 2. For functions of the form y = ax, where 0 < a < 1 • y =ax is a decreasing function • The asymptote is the x-axis or line y =0. • The domain is the set of real number • The range is y/y > 0. 18

• The smaller is the base the steeper is the graph.3. For f(x) = B(ax), the y – Intercept of the graph is multiplied B times and the asymptote is the line y = 0 or the x - axis.4. For f(x) = B(ax) + c , c is the asymptote of the curve and the y – intercept is translated c units from f(x) = B(ax) .What have you LearnedAnswer the following:1. Which of the following statements describes an exponential growth?a. A population grows 5% every year.b. The cost of an apple per kilogram.c. The area of a square of side s.d. The height of a person with respect to his age.2. Which of the following table of values describes an exponential function?a. c. x -2 -1 0 1 2 y9 6 569 x12345 y13579b. d. x -2 -1 0 1 2 y 1 1 139 x0123 4 93 y 2 5 8 11 143. In the function y = 4(3x) – 1, the graph is asymptotic toa. y = 1b. y-axisc. y = -1d. x - axis4. Which function is increasing?a. y = -2xb. y = 2( 1 x) 2c. y = 3(4-x)d. y = -4(2-x) 19

5. What is the y-intercept of the function y = 4(2x) – 2?a. 2b. –2c. 4d. 1From the graph, Y 6 5 4 3 2 1X0 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3determine the:6. y-intercept7. asymptote8. Trend of the graph9. Which is a decreasing function?a. y = 7x – 2b. y = 7 -x + 2c. y = 3(5x) –5d. y = 5(3x) + 310. What function is the reflection of y = 10x with respect to the y-axis? 20

Answer keyHow much do you know 1. d 2. d 3. c 4. 1 5. increasing 6. -2 7. d 8. d 9. a 10. cTry this outLesson 1 1. Exponential growth 2. Exponential decay 3. Exponential growth 4. Exponential growth 5. Exponential decay 6. Exponential decay 7. Exponential growth 8. Exponential decay 9. Exponential growth 10. Exponential growthLesson 2 1. Quadratic 2. Exponential 3. Linear 4. Exponential 5. Exponential 6. Linear 7. Quadratic 8. Exponential 9. Linear 10. Exponential 21

Lesson 31. Table of values x y = 1.5x y = 2.5x y = 3x .037 -3 .296 .064 .111 .33 -2 .444 .16 1 -1 .667 .4 3 9 01 1 27 1 1.5 2.5 2 2.25 6.25 3 3.375 15.625 The graph Y X 17 -4 -3 -2 16The Properties 15 14 13 y = 2.5x y = 3x 12 y = 1.5x 11 1234 10 9 8 7 6 5 4 3 2 1 0 -1 -1 0 -2 -3 Function Domain Range Y- Asymptote Trend1. y = 1.5x y/y > 0 intercept Increasing2. y = 2.5x Set of y/y > 0 x – axis or3. y = 3x Real nos. y/y > 0 I line y = 0 Increasing x – axis or Increasing Set of 1 line y = 0 Real nos. x – axis or 1 line y = 0 Set of Real nos. 22

B. 1. y = 3x 2. y = 2.5x 3. y = 3.5x 4. y = 4x 5. y = 4.2x 6. y = 9x 7 y = 5x 8. y = 8x 9. y = 6x 10. y = 5xLesson 4A. Y1. y = 3x and y = 1 x 29 3 28Table of Values 27x y = 3x y = 1 x y= 1x 26 y = 3x 25 3 3 24 123 23 22-3 1 21 27 27 20 19-2 1 9 18 9 17 16 1 3 15-1 14 3 1301 1 12 1 11 313 10 9 8 129 9 7 63 27 1 5 27 4 3 2 X 1 0 -4 -3 -2 -1 -1 0 4 -2 -3 23

2. If y = 4x and y = 4-xTable of values x -3 -2 -1 0 1 2 3 y = 4x .015 .62 .25 1 4 16 64 y = 4-x 64 16 4 1 .25 .62 .015 The graph y 17 16 15 y = 4-x 14 y = 4x 13 12 11 10 9 8 7 6 5 4 3 2 1x0 -4 -3 -2 -1 -1 0 1 2 3 4 -2 -3B.1. increasing2. decreasing3. increasing4. increasing5. decreasing6. decreasing7. increasing8. increasing9. increasing10.decreasing 24

C.Function Domain Range y- Asymptote Trend y/y > 0 intercept decreasing1. f(x)= 7-x Set of Real y/y > 0 y =0 or the x Increasing2. f(x) = 7x Nos. y/y > 0 1 - axis y/y > 0 Set of Real y/y > 0 1 y =0 or the x Nos. - axis y/y > 03. f(x) = 1 x Set of Real y/y > 0 1 y =0 or the x decreasing Nos. y/y > 0 - axis 5 y/y > 0 Set of Real 1 y =0 or the x decreasing4. f(x) = 5-x Nos. y/y > 0 - axis5. f(x) = 4 x Set of Real 1 y =0 or the x Increasing Nos. - axis 3 Set of Real 1 y =0 or the x decreasing6. f(x) = 2 x Nos. - axis 3 Set of Real 1 y =0 or the x Increasing Nos. - axis7. f(x) = 9x Set of Real 1 y =0 or the x decreasing8. f(x) = 9-x Nos. - axis9. f(x) = 3 x Set of Real 1 y =0 or the x Increasing Nos. - axis 2 Set of Real 1 y =0 or the x decreasing10.f(x) = 3 -x Nos. - axis 2Lesson 5 Y 1. y = 2(2x) 17 16 x y = 2(2x) -3 0.25 15 -2 0.5 14 -1 1 02 13 14 12 28 11 3 16 10 9 8 7 6 5 4 3 2 1 X0 -1 -4 -3 -2 -1 -2 0 1 2 3 4 -3 25

2. y = 3x – 4 Y x y = 3x - 4 6 -3 -3.96 -2 -3.89 5 -1 -3.67 0 -3 4 1 -1 25 3 3 23 2 B. 1. 1 1 2. 1 3. 2 X 4. 3 5. 5 0 6. 2 -5 -4 -3 -2 -1 0 1 2 3 7. 4 8. 8 -1 9. 4 10. -2 -2 C. -3 1. y = 0 2. y = 0 -4 3. y = 0 4. y = 0 5. y = o 6. -2 7. 3 8. -5 9. 2 10. -3What have you Learned 1. a 2. b 3. c 4. b 5. 2 6. -2 7. y = -3 8. increasing 9. b 10. y = 10-x 26

Module 1 Linear Functions What this module is about This module is about a special type of relation called Linear Functions. Inyour study of functions, you learned about relations of quantities in real lifesituations. This time you will concentrate your study on first degree functions,how it is represented through equations and graphs. What you are expected to learn This module is designed for you to: 1. define the linear function f(x) = mx + b. 2. rewrite the standard form Ax + By = C to the slope – intercept form f(x) = mx + b and vice versa. 3. draw the graph of a linear function given the following : • any two points • x and y – intercept • slope and one point • slope and y- intercept How much do you know1. Which of the following functions is linear? a. f(x) = 3x –7 b. f(x) = x( 2 – x) c. f(x) = 2 (6 – 9x) 3 d. 5y = 5x2 – 5 e. 8x – 2y = 42. The graph of 2x + 3 is a __________.3. In the equation y = mx + b, m represents the __________ of the line.

4. In the equation y = 3x + 2, 2 represents the __________ of the line.5. Write 9x – 3y + 6 = 0 in the form y = mx + b.6. The equation y = 3x – 2 if change to Ax + By = C is equal toa. 3x – y = 2 b. 3x + y = 2 c. 3x + y = -2 d. 3x - y = - 2Draw the graph of a linear function given the following:7. Two points: P (0, 0), Q (4, 4)8. Slope (m) = 3; point A (2, 1)9. x – intercept = 3; y- intercept = -1 2

10. Slope (m) = 2, y – intercept = -2 What you will do Lesson 1 Define Linear function f(x) = mx + b A function is a linear function if and only if its equation can be written inthe form y = mx + b or f(x) = mx + b, where m is the slope and b is the y-interceptof the line. The graph is a non-vertical straight line.Examples:1. Is y + 2x = 2 a linear function?Solve for y: y + 2x = 2 The equation is in the form ax + by = c.y + 2x – 2x = -2x + 2 Add – 2x to both sides of the equation. The equation is now in the form y = mx + b y = - 2x + 2 or f(x) = mx + b. or f(x) = -2x + 2 Since the equation can be written in the form y = mx + b or f(x) = mx + b,where m = -2 and b = 2. Then, y + 2x = 2 is a linear function.2. Is x2 – y + 1 = 0 a linear function?Solve for y:x2 – y + 1 = 0 -y = - x2 – 1 3

y = x2 + 1 Since you cannot write the equation in the form y = mx + b. Then, x2 – y + 1= 0 is not a linear function.3. Is y(x + 6) = x(y + 3) a linear function?Solve for y:y(x + 6) = x(y + 3) Apply distributive property of multiplication.xy + 6y = xy + 3x By addition property, add – xy to both sides of the equations 6y = 3x y= 1x 2 After simplifying, the equation is linear since it can be written in the form y= mx + b, where m = 1 and b = 0. 2Try this outA. Which of the following equation is a linear equation?1. y = 2x2. y = - 1 x + 4 53. f(x) = - 5 x 24. x = -75. xy = 106. 7y = 07. f(x) = x + 6 38. f(x) = 2 x+ 3 79. y = 1 + x 310. f(x) = − 2 3B. Determine whether the relation is linear.1. x(y + 3) = 02. 2y + 2x = 6 4

3. 5x – 2y + 6 = 04. 2(x-y) = 3( x + y)5. 2(x + y) = 5 (y + 1) Lesson 2 Given a Linear Function Ax +By = C, Rewrite in the Form f(x) = mx + b, a Slope-Intercept Form and Vice Versa.. The standard form of linear equation Ax + By = C can be transformed to alinear function f(x) = mx +b called slope - intercept form, where m is the slopeand b is the y-intercept and vice versa.Example 1: Write 3y = -2x – 6 in the form f(x)= mx +b. Give the value of m and ba. 3y = -2x – 6 b. 2x – 5y = 10 3 =-2x– 6 33 3 – 5y = -2x + 10 y =-2x -2 3 − 5 y = − 2 x + 10 −5 −5 −5or f(x) = - 2 x - 2 y = 2x -2 3 5m = - 2 , b = -2 or f(x) = 2 x - 2 3 5 m = 2 , b = -2 5Example 2: Write y = 2 x - 2 in the form Ax + By = C. 5y = 2x -2 55[y = 2 x - 2] multiply by the LCD 55y = 2x – 10-2x + 5y = -10-1[-2x + 5y = -10] multiply by -1.2x – 5y = 10 5

Try this outA. Write the following in the form y = mx + b. 1. 4y + 12x = 20 2. 2 x + 9y – 8 = 10 3. 4y + x + 2 = 0 4. 3x – y = 0 5. 8x – 2y = 6 6. 3x – 4y = 8 7. y + 5x – 3 = 0 8. 2y – 6x + 10 = 0 9. 5x + y = 3 10. –y + 2x – 7 = 0B. Write the following linear equation in the form Ax + By = C. 1. y = -2x + 3 2. y = 3x – 1 3. y = 1 x + 3 2 4. y = 2x – 2 5. y = 2x + 1 2 6. y = 9 x + 1 24 7. y = - 7 x – 1 2 8. y = -3x + 5 9. y = 1 x – 2 4 10.y = - 3 x – 3 2 Lesson 3 Draw the Graph of a Linear Function Given Two Points The graph of a linear function is a straight line. In Geometry, you learnedhow to graph by connecting points. In this section you will learn how to graphlinear functions and determine its slope using the following conditions: A. Given any two points Two points determine a straight line, this is a statement in geometry where you can apply to graph linear functions. 6

Example: Draw the graph of a linear function passing through points (1, 2) and (2, 4). a. First locate the two points b. Then connect the two points. The graph of the linear function will look like the figure below. • (2,4) • (1,2) B. Given the x and y – intercepts: Another way of graphing a linear function is through the points where the graph crosses the x and y axes. This condition also uses two points. The point at which a line crosses the y-axis has an x coordinate of 0 called y-intercept. While, the point at which the line crosses the x-axis has y- coordinate of 0 called x-intercept. Example: 1. x - intercept = -3 ; y - intercept = 6 The y – intercept is the y value at point (0, 6). Here the y-intercept is 6 The x – intercept is the x value at point (-3,0). Here the x intercept is -3. 7

Try this outA. Draw the graph of the linear function that passes through the given points.1. (3, -1) and (1, -3) 4. (0, 0) and (2, -2)2. (0, -2) and (-2, 1) 5. (-1, 2) and (3, -2)3. (3, 4) and (-4, -3) 6, (-3 , - 3) and (2, 1) 8

B. Draw the graph of the linear function whose x and y –intercepts are given.1. x- intercept = 1; y – intercept = 4 4. x – intercept = - 3; y – intercept = 22. x-intercept = -3; y-intercept = -2 5. x-intercept = 2; y-intercept = -33. x-intercept = 2; y-intercept = -2 6. x-intercept = -4; y-intercept = -3 9

Lesson 4Draw the Graph of a Linear Function Given the Slope and a PointA. Given the slope m and y – intercept b. The slope m determines the steepness of a line while the y – intercept isthe y value of the point (0, b) where the graph touches the y-axis. The slope is simply m = rise runExample 1: In the graph below, the y – intercept is -2 and the slope is 8 or 4. 2 To graph, start with the y-intercept, and then rise 8 and run 2. These would connect the points (0,-2) and (2, 6) Notice that the slope can be computed using the two points in the formula.y - int m = rise = y2 − y1 run x2 − x1 =8 = 6 - -2 = 8 = 4 2-0 2 = 2 The line rises to the right when the slope is positive.Example 2: Graph the linear function whose y-intercept = -2 and slope (m)= − 3 2 From the y – intercept, at (0, -2) rise 3 and run 2 units to the left. This would connect points (0, -2) and (-2, 1)(-2, 1) 2y-int 3 This time the direction of the graph goes down to the right because the • (0,-2) slope is negative. 10

Try this outA. Draw the graph given the slope and passing through the given point.1. m = 4 ; P(-2, -3) 4. m = 2; P(2, 2) 32. m = 3; P(0, 2) 5. m = -3; P(-2, 3) 23. m = -3 ; P(-1, 1) 6. m = -3; P(1, -3) 5 11

B. Draw the graph with the indicated slope and passing through the given y- intercept.1. m = -5; y-intercept = -3 4. m = 2; y-intercept = -3 32. m = -1; y-intercept = 3 5. m = 1; y-intercept = -2 2 23. m = 3; y-intercept = 1 6. m = -4; y-intercept = -4 4 12

Let’s summarize 1. The standard form of linear equation is ax + by = c, where a, b, and c are real numbers, and a and b are not both zero. 2. Linear function can be defined by f(x) = mx + b, where m is the slope of the line and b is the y – intercept. 3. The graph of a linear function is a straight line. 4. The graph can be drawn if the following are given: • any two points • x and y- intercept • slope and any one point • slope and y-intercept What have you learnedA. Which of the following equations/functions is linear? 1. f(x) = -x + 3 2. f(x) = x(x – 3) 3. y = 2x2 – 3x + 1 4. y = 1 (4 + 6x) 2 5. y = - 4xB. Write each equation in the standard form. 1. y = 5 x + 1 8 2. x = 1 y - 4 3 3. y = − 2 x + 14 3 13

C. Write each equation in the slope-intercept form. 1. 5x + 3y = 9 2. 3x – 2y = 12 3. x + 2y = 7D. Draw the graph of linear functions given the following:. 1. Two points (1, 2), (2, 4)2. slope (m) = -2, passes through (-1, 3) 14

3. x- intercept = 4, y-intercept = 34. slope = 4 , y-intercept = -3 3 15

Answer KeyHow much do you knowA. 1. a. Linear b not linear c. linear d. not linear e. linear2. straight line3. slope4. y-intercept5. y = 3x + 26. a7. Two points: P(0, 0); Q(4, 4) Q: (4,4) • P:(0,0) •8. Slope (m) = 3; point A (2,1) A:(2,1) • 16

9. x- intercept = 3; y –intercept = -1 (3,0) • • (0,-1)10. Slope (m) = 2; y-intercept = -2 (0,-2) •Try this out 6. not linear 7. linearLesson 1 8. linear 9. linear A. 10. not linear 1. linear 2. linear 3. linear 4. not linear 5. not linear B. 1. not linear 2. linear 3. linear 4. linear 5. linear 17

Lesson 2 6. y = 3x - 2 4 A. 1. y = -3x + 5 7. y = -5x + 3 2. y = -2x + 2 9 8. y = 3x – 5 3. y = -1x – 1 9. y = -5x + 3 42 10. y = 2x - 7 4. y = 3x 6. 18x – 4y = -1 5. y = 4x – 3 7. 7x + 2y = -2 8. 3x - y = 5 B. 9. x – 4y = 8 1. 2x + y = 3 10. 3x + 2y = -6 2. 3x – y = 1 3. x – 2y = -6 4. (0, 0), (2, -2) 4. 2x – y = 2 5. 4x – 2y = -1Lesson 3 A 1. (3, -1), (1, -3) • (3,-1) (0,0) • • (2,-2) (1,-3) •2. (0, -2), (-2,1) 5. (-1, 2), (3, -2) 18

3. (3, 4), (4, -3) 6. (-3, -3), (2, 1)B.5. x-intercept = 1, y-intercept = 4 4. x-intercept = -3, y-intercept = 22. x-intercept = -3 ; y-intercept = -2 5. x-intercept = 2; y-intercept = -3 19

3. x-intercept = 2; y-intercept = 2 6. x-intercept = -4; y-intercept = -3Lesson 4 4. m= 2, P(2, 2) A. 1. m = 4, P(-2, -3) 32. m = 3, P(0, 2) 5. m= -3, P(-2, 3) 2 20

3. m= -3; P(1, -1) 6. m= -3; P(1, -3) 5B. 4. m = 2, y-intercept = -31. m = -5, y-intercept = -3 32. m= -1, y-intercept = 3 5. m= 1, y-intercept= -2 2 2 21

3. m= 3, y-intercept = 1 6. m= -4; y-intercept = -4 4What have you learnedA. 1. Linear 2. not linear 3. not linear 4. Linear 5. LinearB. 1. 5x – 8y = 8 2. 3x – y = -12 3. 2x + 3y = 42C. 1. y = − 5 x + 3 3 2. y = 3 x – 6 2 3. y = - 1 x + 7 22 22

D. 1. (1, 2), (2, 4) 3. x-intercept = 4, y-intercept = 32. slope (m) = -2, passes through (-1, 3) 4. slope = 4, y-intercept = -3 3 23